Power Reduction Formulas/Hyperbolic Cosine Squared

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Theorem

$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$

where $\cosh$ denotes hyperbolic cosine.


Proof 1

\(\displaystyle 2 \cosh^2 x - 1\) \(=\) \(\displaystyle \cosh 2 x\) Double Angle Formula for Hyperbolic Cosine:Corollary 1
\(\displaystyle \cosh^2 x\) \(=\) \(\displaystyle \frac {\cosh 2 x + 1} 2\) solving for $\cosh^2 x$

$\blacksquare$


Proof 2

\(\displaystyle \cosh^2 x\) \(=\) \(\displaystyle \frac 1 4 \paren {e^x + e^{-x} }^2\) Definition of Hyperbolic Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{2 x} + e^{-2 x} + 2} 4\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\cosh 2 x + 1} 2\) Definition of Hyperbolic Cosine

$\blacksquare$


Proof 3

\(\displaystyle \cosh^2 x\) \(=\) \(\displaystyle \cos^2 i x\) Hyperbolic Cosine in terms of Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {\map \cos {2 i x} + 1} 2\) Square of Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {\cosh 2 x + 1} 2\) Hyperbolic Cosine in terms of Cosine

$\blacksquare$


Also see


Sources